Degenkolb's Argos-Shimano team-mate Luka Mezgec hit the tarmac on a newly painted zebra crossing coming out of the final bend of the 203km stage from Cosenza - leaving Italy's Marco Canola (Bardiani Valvole) alone on the front inside the final kilometre.
But Degenkolb avoided the ensuing melee and managed to reel back Canola inside the closing 200 metres to take the first Giro d'Italia stage victory of his career.
Spain's Angel Vicioso (Katusha) was second and Dutchman Paul Maertens (Blanco) took third as a splintered pack crossed the line in drips and drabs after another eventful day on the 96th edition of Italy's premiere stage race.
"My lead-out man crashed and there was just one guy in front," said the moustachioed Degenkolb after his first win of the season. "I looked back and saw just [Elia] Viviani behind and so I went full gas to catch the rider from Bardiani."
Despite the crash by his Slovenian team-mate Mezgec - which caused Degenkolb to swerve and clip out of the pedals - the 24-year-old sprinter was quick to thank his Argos-Shimano team for their hard work during another wet stage on the Giro.
"It was a great job by my team. We controlled the race and had the confidence to lead the chase. It was a great day for Argos-Shimano," he said, adding: "I had to suffer a lot to get to the finish."
Italian veteran Luca Paolini (Katusha) - the stage three winner - finished in the main pack to retain his pink jersey as race leader.
Race favourites Vincenzo Nibali (Astana), Ryder Hesjedal (Garmin-Sharp) and Bradley Wiggins (Team Sky) finished safely to stay in fourth, fifth and sixth place respectively. The Italian trails his compatriot Paolini by 31 seconds while last year's winners of the Giro and Tour are both a further three seconds in arrears.
Wiggins' Sky team-mate Rigoberto Uran retained his second place on GC despite puncturing 14km from the finish. The Colombian managed to fight back into the peloton in time for the dramatic finale - and stays 17 seconds down on Paolini. Spain's Benat Intxausti (Movistar) is fifth at 26 seconds.
Six riders broke clear of the peloton after just six sunny opening kilometres. Tomas Gil (Androni Giocattoli), Stefano Pirazzi (Bardiani Valvole), Alan Marangoni (Cannondale), Ricardo Mestre (Euskaltel), Brian Bulgac (Lotto Belisol) and Rafael Andriato (Vini Fantini) built up a maximum lead of nine minutes ahead of the first of two fourth category climbs.
Italy's Pirazzi took maximum points over the first climb before dropping back to the peloton to give the break a chance of survival: trailing Paolini by just 3:06 in the overall standings, Pirazzi was clearly marked out as a danger man by Katusha.
As the race edged along the Gulf of Tarento the weather took a turn for the worse, with the sun and blue sky replaced with grey clouds and torrential rain. At the finish in Matera there was so much water that the closing straight had become a river and pieces of media equipment were being washed away.
The remaining five leaders were reeled in on the last climb of the day, the Cat.4 Montescaglioso, 22km from the finish.
Sprinters Mark Cavendish (Omega Pharma-Quick Step) and Matt Goss (Orica-GreenEdge) were both tailed off on the short and sharp ascent as numerous riders tried their luck off the front.
Frenchman Hubert Dupont (Ag2R-La Mondiale) had a dig 6km from the finish but was swallowed up by the peloton with 3km to go.
Although the Matera monsoon had long since stopped and the sun had returned, the roads into the town were still wet as the pack roared into the picturesque settlement best known for its ancient village of hilltop caves and stone houses - the Sassi.
With Canola entering the final bend in pole position to set things up for team-mates Sacha Modolo and Tuesday's winner, Enrico Battaglin, Argos-Shimano's Mezgec had similar intentions for Degenkolb.
But the Slovenian lost his front wheel and slipped into the barriers at top speed, bringing down numerous riders with him and causing a massive split in the peloton.
Degenkolb, winner of five stages on last year's Vuelta, showed remarkable bike-handling skills to avoid hitting the deck - and after one look back to survey the devastation in his wake, he set off in pursuit of the unlikely lone leader.
Canola gave it his best shot but was no match for the power of Degenkolb who roared across the line to open up his Giro account in style.
Degenkolb will be in the red points jersey when the race continues on Thursday with the flat 169km stage six along the Adriatic Coast from Mola di Bari to Margherita di Savoia in Puglia.